cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A218098 Number of transitive reflexive early confluent binary relations R on n labeled elements with max_{x}(|{y : xRy}|) = 8.

Original entry on oeis.org

545835, 27733869, 1173919350, 47488375440, 1933688266686, 81009491387682, 3527548086703069, 160415345420268510, 7631859877504516225, 379961855272982538127, 19785139747357478264082, 1076480694153554931849504, 61126131119735946242652270
Offset: 8

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

References

  • A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

Crossrefs

Column k=8 of A135313.

Programs

  • Maple
    t:= proc(k) option remember; `if`(k<0, 0,
      unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
    end:
egf:= t(8)(x)-t(7)(x):
a:= n-> n!* coeff(series(egf, x, n+1), x, n):
seq(a(n), n=8..22);
  • Mathematica
    m = 8; t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]] ; egf = t[m][x]-t[m-1][x]; a[n_] := n!*Coefficient[Series[egf, {x, 0, n+1}], x, n]; Table[a[n], {n, m, 22}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

Formula

E.g.f.: t_8(x)-t_7(x), with t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) if k>=0 and t_k(x) = 0 else.
a(n) = A210916(n) - A210915(n).

A218099 Number of transitive reflexive early confluent binary relations R on n labeled elements with max_{x}(|{y : xRy}|) = 9.

Original entry on oeis.org

7087261, 451238935, 22913136730, 1087116745385, 51075201835515, 2437976801668408, 119752042470064290, 6093096859120003590, 322215964319093498225, 17735784941946000072572, 1016521929886047797022408, 60650840653136697085038930, 3764766650086543657134295955
Offset: 9

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

References

  • A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

Crossrefs

Column k=9 of A135313.

Programs

  • Maple
    t:= proc(k) option remember; `if`(k<0, 0,
      unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
    end:
egf:= t(9)(x)-t(8)(x):
a:= n-> n!* coeff(series(egf, x, n+1), x, n):
seq(a(n), n=9..22);
  • Mathematica
    m = 9; t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]] ; egf = t[m][x]-t[m-1][x]; a[n_] := n!*Coefficient[Series[egf, {x, 0, n+1}], x, n]; Table[a[n], {n, m, 22}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

Formula

E.g.f.: t_9(x)-t_8(x), with t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) if k>=0 and t_k(x) = 0 else.
a(n) = A210917(n) - A210916(n).

A218103 Number of transitive reflexive early confluent binary relations R on n+3 labeled elements with max_{x}(|{y : xRy}|) = n.

Original entry on oeis.org

0, 1, 310, 12075, 267715, 5287506, 105494886, 2185028130, 47488375440, 1087116745385, 26234041133443, 666937354457829, 17839235553096685, 501241620987647540, 14769149279798467900, 455566464561064320948, 14685947990441112405726, 493969048893703131221475
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Links

Programs

  • Maple
    t:= proc(k) option remember; `if`(k<0, 0,
      unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
    end:
tt:= proc(k) option remember; unapply((t(k)-t(k-1))(x), x) end:
T:= proc(n, k) option remember;
      coeff(series(tt(k)(x), x, n+1), x, n) *n!
    end:
a:= n-> T(n+3,n):
seq(a(n), n=0..20);
  • Mathematica
    m = 3; f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]]; (* t = A135302 *) t[0, 0] = 1; t[, 0] = 0; t[n, k_] := t[n, k] = SeriesCoefficient[f[k, x], {x, 0, n}]*n!; a[0] = 0; a[n_] := t[n+m, n]-t[n+m, n-1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 14 2014 *)

Formula

a(n) = A135313(n+3,n).
a(n) ~ n! * n^6 / (96 * log(2)^(n+4)). - Vaclav Kotesovec, Nov 20 2021
Conjecture: For fixed k>=0, A135313(n+k,n) ~ n! * n^(2*k) / (2^(k+1) * k! * log(2)^(n+k+1)). - Vaclav Kotesovec, Nov 20 2021

A218104 Number of transitive reflexive early confluent binary relations R on n+4 labeled elements with max_{x}(|{y : xRy}|) = n.

Original entry on oeis.org

0, 1, 1821, 141533, 4798983, 124878033, 3068829477, 75967708311, 1933688266686, 51075201835515, 1405508547112670, 40356644902123914, 1209368372802130814, 37806870603888974350, 1231961629420423620918, 41802174277488971170242, 1475352032068521550599837
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Links

Programs

  • Maple
    t:= proc(k) option remember; `if`(k<0, 0, unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x)) end: tt:= proc(k) option remember; unapply((t(k)-t(k-1))(x), x) end: T:= proc(n, k) option remember; coeff(series(tt(k)(x), x, n+1), x, n) *n! end:
a:= n-> T(n+4,n): seq(a(n), n=0..20);
  • Mathematica
    m = 4; f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]]; (* t = A135302 *) t[0, 0] = 1; t[, 0] = 0; t[n, k_] := t[n, k] = SeriesCoefficient[f[k, x], {x, 0, n}]*n!; a[0] = 0; a[n_] := t[n+m, n]-t[n+m, n-1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 14 2014 *)

Formula

a(n) = A135313(n+4,n).
a(n) ~ n! * n^8 / (768 * log(2)^(n+5)). - Vaclav Kotesovec, Nov 20 2021

A218105 Number of transitive reflexive early confluent binary relations R on n+5 labeled elements with max_{x}(|{y : xRy}|) = n.

Original entry on oeis.org

0, 1, 11592, 1812216, 92374107, 3151808478, 94494907584, 2755081426548, 81009491387682, 2437976801668408, 75638497021149062, 2427804103875438288, 80751743315656443940, 2784897386029995089700, 99580133563729334883624, 3690405873805797826482120
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Links

Crossrefs

Cf. A135313.

Programs

  • Maple
    t:= proc(k) option remember; `if` (k<0, 0, unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x)) end: tt:= proc(k) option remember; unapply((t(k)-t(k-1))(x), x) end: T:= proc(n, k) option remember; coeff(series(tt(k)(x), x, n+1), x, n) *n! end:
a:= n-> T(n+5,n): seq(a(n), n=0..20);
  • Mathematica
    m = 5; f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]]; (* t = A135302 *) t[0, 0] = 1; t[, 0] = 0; t[n, k_] := t[n, k] = SeriesCoefficient[f[k, x], {x, 0, n}]*n!; a[0] = 0; a[n_] := t[n+m, n]-t[n+m, n-1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 14 2014 *)

Formula

a(n) = A135313(n+5,n).

A218106 Number of transitive reflexive early confluent binary relations R on n+6 labeled elements with max_{x}(|{y : xRy}|) = n.

Original entry on oeis.org

0, 1, 80963, 25188019, 1913052805, 84934607175, 3085918099231, 104970367609107, 3527548086703069, 119752042470064290, 4150321205365373610, 147666165472551221730, 5409628424337030402002, 204363410596110256258446, 7966805463258438079563650
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Links

Programs

  • Maple
    t:= proc(k) option remember; `if` (k<0, 0, unapply (exp (add (x^m/m! *t(k-m)(x), m=1..k)), x)) end: tt:= proc(k) option remember; unapply ((t(k)-t(k-1))(x), x) end: T:= proc(n, k) option remember; coeff (series (tt(k)(x), x, n+1), x, n) *n! end:
a:= n-> T(n+6,n): seq (a(n), n=0..20);
  • Mathematica
    m = 6; f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]]; (* t = A135302 *) t[0, 0] = 1; t[, 0] = 0; t[n, k_] := t[n, k] = SeriesCoefficient[f[k, x], {x, 0, n}]*n!; a[0] = 0; a[n_] := t[n+m, n]-t[n+m, n-1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 14 2014 *)

Formula

a(n) = A135313(n+6,n).

A218107 Number of transitive reflexive early confluent binary relations R on n+7 labeled elements with max_{x}(|{y : xRy}|) = n.

Original entry on oeis.org

0, 1, 608832, 378725365, 42483670075, 2440299822081, 106818340013957, 4210378306984993, 160415345420268510, 6093096859120003590, 234104217274598884642, 9167943015777908270142, 367520396335132750893274, 15117877192137817244318510, 638973577773301815522889410
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Links

Programs

  • Maple
    t:= proc(k) option remember; `if` (k<0, 0, unapply (exp (add (x^m/m! *t(k-m)(x), m=1..k)), x)) end: tt:= proc(k) option remember; unapply ((t(k)-t(k-1))(x), x) end: T:= proc(n, k) option remember; coeff (series (tt(k)(x), x, n+1), x, n) *n! end:
a:= n-> T(n+7,n): seq (a(n), n=0..20);
  • Mathematica
    m = 7; f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]]; (* t = A135302 *) t[0, 0] = 1; t[, 0] = 0; t[n, k_] := t[n, k] = SeriesCoefficient[f[k, x], {x, 0, n}]*n!; a[0] = 0; a[n_] := t[n+m, n]-t[n+m, n-1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 14 2014 *)

Formula

a(n) = A135313(n+7,n).

A218108 Number of transitive reflexive early confluent binary relations R on n+8 labeled elements with max_{x}(|{y : xRy}|) = n.

Original entry on oeis.org

0, 1, 4910785, 6135529675, 1008618127825, 74564772630777, 3913397076494906, 177779119899659850, 7631859877504516225, 322215964319093498225, 13636766011245325587353, 584294217357391235758011, 25488316708898114509899955, 1135731969645865474902932115
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Links

Programs

  • Maple
    t:= proc(k) option remember; `if` (k<0, 0, unapply (exp (add (x^m/m! *t(k-m)(x), m=1..k)), x)) end: tt:= proc(k) option remember; unapply ((t(k)-t(k-1))(x), x) end: T:= proc(n, k) option remember; coeff (series (tt(k)(x), x, n+1), x, n) *n! end:
a:= n-> T(n+8,n): seq (a(n), n=0..20);
  • Mathematica
    m = 8; f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]]; (* t = A135302 *) t[0, 0] = 1; t[, 0] = 0; t[n, k_] := t[n, k] = SeriesCoefficient[f[k, x], {x, 0, n}]*n!; a[0] = 0; a[n_] := t[n+m, n]-t[n+m, n-1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 14 2014 *)

Formula

a(n) = A135313(n+8,n).

A218109 Number of transitive reflexive early confluent binary relations R on n+9 labeled elements with max_{x}(|{y : xRy}|) = n.

Original entry on oeis.org

0, 1, 42159238, 106586385708, 25519311555595, 2416548374532292, 151442406160585540, 7894403946290257968, 379961855272982538127, 17735784941946000072572, 822369813313954835099742, 38353581871007817965010668, 1811813065380635747237663856
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Links

Crossrefs

Cf. A135313.

Programs

  • Maple
    t:= proc(k) option remember; `if`(k<0, 0, unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x)) end: tt:= proc(k) option remember; unapply((t(k)-t(k-1))(x), x) end: T:= proc(n, k) option remember; coeff(series(tt(k)(x), x, n+1), x, n) *n! end:
a:= n-> T(n+9,n): seq(a(n), n=0..20);
  • Mathematica
    m = 9; f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]]; (* t = A135302 *) t[0, 0] = 1; t[, 0] = 0; t[n, k_] := t[n, k] = SeriesCoefficient[f[k, x], {x, 0, n}]*n!; a[0] = 0; a[n_] := t[n+m, n]-t[n+m, n-1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 14 2014 *)

Formula

a(n) = A135313(n+9,n).

A218110 Number of transitive reflexive early confluent binary relations R on n+10 labeled elements with max_{x}(|{y : xRy}|) = n.

Original entry on oeis.org

0, 1, 383478987, 1976799958367, 686016182577453, 82847673438018762, 6177363078563029080, 368141001305925742232, 19785139747357478264082, 1016521929886047797022408, 51404873131596488549863350, 2597923441011463423121994276, 132340384137811145863910654038
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Links

Programs

  • Maple
    t:= proc(k) option remember; `if` (k<0, 0, unapply (exp (add (x^m/m! *t(k-m)(x), m=1..k)), x)) end: tt:= proc(k) option remember; unapply ((t(k)-t(k-1))(x), x) end: T:= proc(n, k) option remember; coeff (series (tt(k)(x), x, n+1), x, n) *n! end:
a:= n-> T(n+10,n): seq (a(n), n=0..20);
  • Mathematica
    m = 10; f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]]; (* t = A135302 *) t[0, 0] = 1; t[, 0] = 0; t[n, k_] := t[n, k] = SeriesCoefficient[f[k, x], {x, 0, n}]*n!; a[0] = 0; a[n_] := t[n+m, n]-t[n+m, n-1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 14 2014 *)

Formula

a(n) = A135313(n+10,n).
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